Utafiti huu unachunguza modified form ya teleparallel gravity inayofafanua gravitation kupitia torsion badala ya spacetime curvature. Watafiti wamependekeza three-term model kwa f(T) function inayotegemea torsion scalar, yenye linear term, inverse-power term na power-law term; kisha wamejaribu kama model hii inaweza kuzalisha radiation, matter na late-time accelerated-expansion eras za Universe kwa kutumia both dynamical-system analysis na observational data.

Three main critical points zimetambuliwa katika phase space ya model. A2 point inawakilisha early-time radiation regime, A3 matter domination, na A1 late-time de Sitter-like accelerated expansion. Katika parameter region ya m < 1 na n > −1 iliyochaguliwa na watafiti, radiation point ilipatikana unstable, matter point ikiwa saddle-like, na dark-energy point ikiwa stable attractor chini ya certain additional conditions. Hivyo model inaweza kutengeneza qualitative cosmological sequence kutoka early radiation era kwenda matter era na kisha accelerated expansion.

Katika observational analysis, 32 Hubble-parameter measurements, Pantheon+SH0ES supernova compilation na DESI DR2 baryon-acoustic-oscillation data zilitathminiwa kwa Markov Chain Monte Carlo method. Katika combined-data analysis, Hubble constant H₀ = 74,135^+0,176_−0,177 km/s/Mpc na present matter density Ω_m0 = 0,315^+0,059_−0,073 zilipatikana. Model results za m = 0,486^+0,319_−0,321 na n = 0,352^+0,123_−0,117 zinaendana na preferred region ya m < 1 na n > −1 katika dynamical analysis.

Model inareproduce Hubble expansion-rate na supernova distance-modulus data kwa namna iliyo karibu na standard ΛCDM model. Hata hivyo, statistical results si za mwelekeo mmoja. Katika combined data, Akaike Information Criterion inaipa model limited advantage, huku Bayesian Information Criterion inayoweka stronger parameter penalty ikiipendelea ΛCDM clearly. Kwa hiyo, study inaonyesha kwamba f(T) model inaweza kuwa observationally viable alternative; lakini haithibitishi kwamba ni superior au correct kuliko standard cosmology.

Main research question ni nini?

Main question ya study ni kama observed accelerated expansion ya Universe inaweza kuelezwa kupitia teleparallel modification ya gravity law badala ya separate dark-energy component. Kwa lengo hili, watafiti wamechagua specific f(T) function inayoweza both kuzalisha theoretically appropriate cosmological eras na kufitiwa kwa current expansion data.

Katika standard general relativity, gravity inafafanuliwa kwa curvature ya spacetime. Katika teleparallel equivalent approach, same classical gravitational physics inaelezwa kupitia Weitzenböck connection yenye zero curvature lakini nonzero torsion. Primary variable kabla ya metric ni four-component frame field inayoitwa tetrad au vierbein.

Model iliyopendekezwa katika study inapanua teleparallel equivalent ya general relativity kwa additional function inayoitwa f(T). Hivyo lengo ni kufanya torsion iongeze new effective energy-density na pressure terms kwenye cosmological dynamics.

Tetrad na torsion zinamaanisha nini katika teleparallel gravity?

Tetrad fields zinaunganisha local flat Lorentz frame katika kila point ya spacetime na general coordinate system. Metric tensor inapatikana kutoka tetrads hivi:

\[ g_{\mu\nu}(x)=\eta_{ij}h^{i}{}_{\mu}(x)h^{j}{}_{\nu}(x) \]

Hapa η_ij ni Minkowski metric, hⁱ_μ ni tetrad components, na g_μν ni spacetime metric.

Torsion tensor ya Weitzenböck connection inaandikwa:

\[ T^{\lambda}{}_{\mu\nu} =h^{\lambda}{}_{i} \left(\partial_{\mu}h^{i}{}_{\nu} -\partial_{\nu}h^{i}{}_{\mu}\right) \]

kwa namna hiyo. Torsion tensor inapima jinsi tetrad field inavyobadilika across spacetime. Katika general relativity gravitational information inabebwa na curvature, huku katika teleparallel approach corresponding information ikiwakilishwa kupitia torsion.

Torsion scalar inapatikana kwa contraction ya superpotential na torsion tensor:

\[ T=S^{\mu\nu}{}_{\rho}T^{\rho}{}_{\mu\nu} \]

Study inatumia “pure tetrad” f(T) formulation inayochukua spin connection kuwa zero. Choice hii inarahisisha mathematical operations; lakini covariance issue inayohusiana na different tetrad na connection choices haikutestwa separately katika study.

Action ya model na field equations zimefafanuliwaje?

Gravitational na matter action imeandikwa hivi:

\[ S=\frac{1}{16\pi G}\int d^{4}x\,e \left[T+f(T)+\mathcal{L}_{m}\right] \]

Hapa e ni determinant ya tetrad matrix, G ni Newton gravitational constant na ℒ_m ni matter Lagrangian density. Action ina both standard linear T term na f(T) term inayobeba modified-gravity effects.

Kwa flat, homogeneous na isotropic Universe, tetrad imechaguliwa kama:

\[ e^{a}{}_{\mu}=\operatorname{diag}(1,a(t),a(t),a(t)) \]

Corresponding flat FLRW line element ni:

\[ ds^{2}=-dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2}) \]

na torsion scalar ni:

\[ T=-6\left(\frac{\dot a}{a}\right)^{2}=-6H^{2} \]

H = ȧ/a ni Hubble parameter na ina inverse-time dimension.

Modified Friedmann equations ni:

\[ 3H^{2}=8\pi G\rho-\frac{f(T)}{2}+Tf_{T} \]

\[ \dot H= -\frac{4\pi G(\rho+p)} {1+f_{T}+2Tf_{TT}} \]

f_T na f_TT ni first na second derivatives za f function kwa T. Term ya 1 + f_T + 2Tf_TT katika denominator inafanya response ya expansion rate kwa matter na pressure kuwa tofauti na standard teleparallel theory.

Proposed f(T) function ni nini?

Watafiti walitumia functional form ifuatayo:

\[ f(T)=\alpha T-\beta u^{-n}+\gamma u^{m} \]

\[ u=\frac{-T}{6}=H^{2} \]

Roles za terms katika model ni hizi:

• αT: Linear additional contribution kwa torsion scalar.
• −βu⁻ⁿ: Term inayotegemea inverse power ya Hubble scale na inaweza kuwa influential hasa kwa low H values.
• γu^m: Term inayotegemea positive au fractional power ya Hubble scale.
• m na n: Exponents zinazoamua jinsi modified-gravity terms zinavyobadilika na expansion rate.
• α, β na γ: Model parameters zinazoamua amplitude ya corresponding terms.

Derivatives za model kwa T ni:

\[ f_{T}=\alpha-\frac{\beta n}{6}u^{-(n+1)} -\frac{\gamma m}{6}u^{m-1} \]

\[ f_{TT}= -\frac{\beta n(n+1)}{36}u^{-(n+2)} +\frac{\gamma m(m-1)}{36}u^{m-2} \]

kama zilivyotolewa. Derivatives hizi zinaingia both effective dark-energy density na cosmological expansion equation.

Effective dark energy imefafanuliwaje?

Additional contributions zinazotokana na f(T) terms zinaandikwa formally kama effective dark-energy component:

\[ \rho_{\mathrm{DE}} =\frac{1}{16\pi G}\left(-f+2Tf_{T}\right) \]

\[ p_{\mathrm{DE}} =-\frac{1}{16\pi G} \left[ -f+Tf_{T} -\frac{2T^{2}f_{TT}} {1+f_{T}+2Tf_{TT}} \right] \]

Corresponding equation-of-state parameter ni:

\[ \omega_{\mathrm{DE}}= -1+ \frac{(f_{T}+2Tf_{TT})(-f+T+2Tf_{T})} {(1+f_{T}+2Tf_{TT})(-f+2Tf_{T})} \]

kwa namna hii. ω_DE = −1 inaonyesha cosmological-constant behavior, values greater than −1 zinaonyesha quintessence-like regime na values less than −1 zinaonyesha phantom-like regime.

Kwa nini cosmological equations zilibadilishwa kuwa dynamical system?

Modified Friedmann equations ni nonlinear na difficult kusolve analytically. Dynamical-system method inabadilisha equations kuwa first-order autonomous system kwa dimensionless variables, kuruhusu long-term behavior ya Universe kuchunguzwa bila kusolve all details za initial conditions.

Katika study, dimensionless variables zifuatazo zimefafanuliwa:

\[ \Omega_{m}=\frac{\kappa^{2}\rho_{m}}{3H^{2}} \]

\[ r=\frac{\kappa^{2}\rho_{r}}{3H^{2}} \]

\[ x=\frac{\beta u^{-n}}{3H^{2}} \left(\frac{1}{2}+n\right) \]

\[ y=-\left[ \frac{\gamma u^{m}}{3H^{2}} \left(\frac{1}{2}-m\right)+\alpha \right] \]

Variables hizi zinatimiza constraint:

\[ \Omega_{m}+r+x+y=1 \]

Hapa r inawakilisha radiation-density parameter, huku x na y zikiwakilisha cosmological contributions za modified torsion terms.

Kwa kuwa x na y zinategemea same Hubble variable, si fully independent. Study inasema kwamba relation:

\[ x^{m-1}(\alpha+y)^{n+1}=\mathrm{sabittir} \]

inaconfine phase space kwenye lower-dimensional submanifold.

Main equations za autonomous system ni zipi?

e-folding number N = ln(a) ikitumiwa, system inakuwa:

\[ x'=-2(n+1)x\frac{\dot H}{H^{2}} \]

\[ y'=2(m-1)(\alpha+y)\frac{\dot H}{H^{2}} \]

\[ r'=-2r\left(2+\frac{\dot H}{H^{2}}\right) \]

Katika uploaded text, third equation ina symbol γ′ badala ya r′; variable definitions na equation structure zinaonyesha kwamba hii ni typographical error.

Dimensionless derivative ya expansion rate ni:

\[ \frac{\dot H}{H^{2}}= \frac{\frac{3}{2}(x+y-1)-\frac{r}{2}} {1+nx-my+\alpha(1-m)} \]

Total equation of state na deceleration parameter zimepewa kama:

\[ \omega_{\mathrm{tot}} =-1- \frac{(x+y-1)-r/3} {1+nx-my+\alpha(1-m)} \]

\[ q=-1+ \frac{r-3(x+y-1)} {2[\alpha-m(\alpha+y)+nx+1]} \]

q < 0 inaonyesha accelerated cosmological expansion, huku q > 0 ikionyesha decelerating expansion.

Critical points tatu zinawakilisha cosmic eras zipi?

Kwa nini A1 dark-energy point ilihitaji special analysis?

Eigenvalues za A₁ point zilipatikana kuwa:

\[ \{0,-4,-3\} \]

Kwa sababu one eigenvalue ni zero, point si hyperbolic na linear stability analysis pekee haitoshi. Kwa hiyo study ilitumia center-manifold theory.

Ingawa sentence moja katika text inatumia word “hyperbolic” kwa A₁, immediately before na after inasema point ni “non-hyperbolic” kwa sababu ya zero eigenvalue na center-manifold analysis inatumika. Mathematical procedure inaonyesha kwamba point ilitreatiwa kama non-hyperbolic critical set.

Katika phase portrait kwenye page 6, nearby trajectories zinaonekana kuconverge toward A₁. Behavior hii inaunga mkono interpretation ya point kama late-time attractor. Hata hivyo, stability condition inayotokana na center-manifold result ni complex inequality inayotegemea α, m, n na b parameter iliyotumika katika transformation. Connection ya b parameter na original phase-space coordinates haijaonyeshwa sufficiently clearly katika text.

Stability ya radiation na matter points ikoje?

Eigenvalues za A₂ radiation point zimetolewa kama:

\[ \{1,-4(m-1),4(n+1)\} \]

Wakati m < 1 na n > −1, point ni unstable kwa sababu positive eigenvalues zipo. Hii ni expected behavior kwa sababu Universe inahitaji kuondoka radiation era na kwenda later stages.

Eigenvalues za A₃ matter point ni:

\[ \{-1,3-3m,3(n+1)\} \]

Kwa m < 1 na n > −1, both positive na negative eigenvalues zipo, kwa hiyo point ni saddle-like. Hivyo matter era inaweza kuwepo temporarily, lakini system hatimaye inaweza kuondoka point hii na kuelekea dark-energy attractor.

Katika phase portrait kwenye page 6, trajectories around A₂ na A₃ zinaonekana kuondoka kwenye points hizi, huku zile around A₁ zikiconverge.

Je, density values katika critical points zinatengeneza physical constraint?

Katika Table II ya study, katika radiation point:

\[ \Omega_{r}=1+\alpha,\qquad \Omega_{\mathrm{DE}}=-\alpha \]

na katika matter point:

\[ \Omega_{m}=1+\alpha,\qquad \Omega_{r}=-\alpha \]

values zimetolewa. Ikiwa density parameters zinatakiwa zisiwe negative, range ya −1 ≤ α ≤ 0 inatokea kwa both points. Kwa upande mwingine, central value ya combined MCMC analysis ni α = 0,210. Ingawa confidence interval pia inajumuisha negative values, central value haipatani directly na all densities katika critical points kuwa positive simultaneously. Study haijadili theoretical–observational tension hii kwa undani.

Cosmological-evolution plots zinaonyesha nini?

Kwenye page 7, Kielelezo 2 kinaonyesha evolution ya equation-of-state, density na deceleration parameters dhidi ya redshift kwa initial values za x₀ = 10⁻¹², y₀ = 0,674 na r₀ = 9,1 × 10⁻²³.

Main results zilizoripotiwa na watafiti ni:

• Katika early era, radiation density ni dominant.
• Baadaye temporary matter-dominated era inatokea.
• Katika late era, effective dark-energy density inakuwa dominant.
• Present values ni roughly Ω_m ≈ 0,3 na Ω_DE ≈ 0,7.
• Matter–radiation equality inatokea roughly katika z ≈ 3387.
• Transition kutoka deceleration kwenda acceleration iko roughly katika z ≈ 0,66.
• Present deceleration parameter ni roughly q₀ ≈ −0,53.
• Present dark-energy equation of state ni roughly ω_DE ≈ −1,015.

ω_DE kuwa slightly below −1 inaonyesha kwamba model inaweza kuonyesha mild phantom-like behavior leo. Katika late era, both total equation of state na dark-energy parameter zinakaribia −1.

Hubble equation kwa observational analysis iliundwaje?

Kwa kutumia condition ya leo H = H₀, β parameter iliamuliwa kwa terms za other model parameters. Hivyo katika MCMC analysis, β haikutumika kama independent free parameter; model ilitestwa kwa:

\[ \theta= (H_{0},\alpha,\gamma,\Omega_{m0},m,n) \]

parameters.

Resulting H(z) equation ni implicit equation; H inaonekana kwenye both sides kwa kila redshift. Kwa hiyo watafiti walitumia numerical solution badala ya analytical solution. H(z) ilisclaeiwa kwa H₀ na kufanywa dimensionless kama:

\[ E(z)=\frac{H(z)}{H_{0}} \]

na condition E(0) = 1 ikatumika.

Ni observational data zipi zilitumika?

Chi-square iliyotumika kwa H(z) data ni:

\[ \chi^{2}_{H(z)} =\sum_{i=1}^{32} \frac{[H_{\mathrm{th}}(z_i)-H_{\mathrm{obs}}(z_i)]^{2}} {\sigma_i^{2}} \]

Supernova distance modulus ni:

\[ \mu_{\mathrm{th}}(z,\theta) =5\log_{10}[d_{L}(z,\theta)]+25 \]

\[ d_{L}(z,\theta) =(1+z)c\int_{0}^{z}\frac{dz'}{H(z',\theta)} \]

na supernova chi-square value ni:

\[ \chi^{2}_{\mathrm{SN}} =\Delta\mu^{T}C^{-1}\Delta\mu \]

kama zilivyofafanuliwa. Kwa DESI pia, difference kati ya observed na theoretical distance ratios ilitathminiwa kwa full covariance matrix.

Total likelihood iliundwa kwa:

\[ \chi^{2}_{\mathrm{toplam}} =\chi^{2}_{H(z)} +\chi^{2}_{\mathrm{DESI}} +\chi^{2}_{\mathrm{SN}} \]

\[ \ln\mathcal{L}=-\frac{1}{2}\chi^{2} \]

expressions. Python-based emcee MCMC package ilitumika kusample posterior distributions.

Numerical results za MCMC analysis ni zipi?

Ranges katika table ni 68-percent, yaani 1σ confidence intervals. Katika all data combinations, central value ya m iko below 1 na central value ya n iko above −1. Kwa hiyo central values ziko katika m < 1 na n > −1 region inayofanya radiation point kuwa unstable na matter point kuwa saddle-like.

Hata hivyo, uncertainty ya γ parameter ni much larger kuliko central value yake. Katika two-dimensional contour plot kwenye page 10, broad na partly flattened confidence regions zinaonekana kati ya γ, α, m na Ω_m0. Appearance hii inaonyesha significant parameter degeneracy na kwamba background-expansion data haziwezi kuconstrain all parameters strongly one by one.

Very narrow H₀ range katika combined analysis imetokana na data combination inayojumuisha Pantheon+SH0ES calibration. Value hii ni posterior result chini ya data na assumptions zilizotumika katika study; haipaswi kutathminiwa kama independent na model-free new measurement ya Hubble constant.

Model ilifiti observational curves vizuri kiasi gani?

Kwenye page 11, katika left panel ya Kielelezo 4, H(z) measurements zinaonyeshwa pamoja na error bars. Red curve ya f(T) model na dashed black curve ya ΛCDM zinaoverlap kwa kiasi kikubwa katika observed redshift range.

Right panel inaonyesha Pantheon+SH0ES distance moduli. f(T) na ΛCDM curves ziko karibu kiasi kwamba karibu hazitofautishiki. Hii inaonyesha kwamba proposed model inaweza kureproduce background expansion history; lakini pia inaonyesha kwamba used data haziwezi kutenganisha strongly physical differences kati ya model na ΛCDM.

AIC na BIC results zinasema nini?

Akaike na Bayesian information criteria zinaweka penalty si kwa minimum chi-square pekee, bali pia kwa number ya free parameters katika model:

\[ \mathrm{AIC}=\chi^{2}_{\min}+2k \]

\[ \mathrm{BIC}=\chi^{2}_{\min}+k\ln N \]

Hapa k inaonyesha number ya parameters na N number ya data points.

Katika individual data groups, positive ΔAIC na ΔBIC values zinafavor standard ΛCDM model. Especially kwa Pantheon+SH0ES, ΔBIC = 29,2362 inaonyesha kwamba Bayesian penalty kwa additional parameters ni high.

Katika combined dataset, ΔAIC = −3,7700 inaonyesha, according to sign definition iliyotumiwa na watafiti, kwamba f(T) model inapata limited advantage kwa AIC. Kwa upande mwingine, ΔBIC = 18,0903 inaonyesha preference ya ΛCDM kwa sababu ya stronger complexity penalty.

Kwa hiyo statistical result haiwezi kusummarizeiwa kama “f(T) model ni superior kuliko ΛCDM”. More accurate conclusion ni kwamba model inaweza kufit observations vizuri; lakini kama improvement ya fit inajustify additional free parameters inategemea information criterion iliyotumika.

Strengths za study ni zipi?

• Inaunganisha theoretical stability analysis na observational parameter estimation katika same model.
• Inawakilisha radiation, matter na dark-energy eras kwa separate critical points.
• Kwa late-time point yenye zero eigenvalue, haitumii linear analysis pekee bali pia center-manifold theory.
• Inachunguza H(z), Pantheon+SH0ES na current DESI DR2 BAO data separately na jointly.
• Inatoa parameter uncertainties kupitia MCMC posteriors.
• Inalinganisha model na ΛCDM si kwa chi-square pekee bali pia kwa AIC na BIC.
• Inaonyesha observational error bars kwa Hubble parameter na distance modulus pamoja na model curves.
• Inakagua consistency kati ya dynamical stability region ya model parameters na observational central values.

Limitations na internal inconsistencies za study ni zipi?

• Study ni preprint ambayo haijapitia peer review.
• Prior distributions, walker count, chain length, burn-in section, acceptance ratio na convergence criteria zilizotumika katika MCMC analysis hazijaelezwa.
• Hakuna reproducible repository link iliyotolewa kwa MCMC code, data-processing scripts na posterior samples.
• γ parameter ina very broad confidence intervals na constrained weakly na background data.
• Model imelinganishwa only na background-expansion data; structure growth, fσ₈, full CMB likelihood au cosmological perturbations hazikutestwa.
• Katika Pantheon+SH0ES analysis, handling ya absolute-magnitude calibration na possible additional nuisance parameters haijaelezwa kwa undani.
• Katika DESI analysis, jinsi sound horizon r_d calculation ilivyofungwa na early-Universe parameters haijafafanuliwa sufficiently.
• Effect ya pure-tetrad na zero-spin-connection choice kwenye results haijachunguzwa.
• Katika description ya A₁ point, kuna textual contradiction kati ya “hyperbolic” na “non-hyperbolic”.
• Katika autonomous-system equation, γ′ inaonekana kuandikwa badala ya r′.
• Katika sentence inayofafanua A₁ densities, Ω_r imeandikwa mara mbili; moja inatarajiwa kuwa Ω_m.
• Relationship ya b parameter iliyotumika katika center-manifold stability condition na physical phase-space variables haijahitimishwa clearly.
• Kuna possible inconsistency kati ya positive-density conditions za critical points na combined-MCMC central value α = 0,210.
• BIC inaipendelea ΛCDM katika all comparisons, including combined data.

Study inaunga mkono nini?

• Inaunga mkono kwamba selected f(T) function inaweza kutengeneza phase space yenye radiation, matter na de Sitter-like acceleration eras.
• Inaonyesha kwamba katika m < 1 na n > −1 region, early radiation point inaweza kuwa unstable na matter point saddle-like.
• Inaonyesha kwamba chini ya certain additional conditions, dark-energy point inaweza kuwa late-time attractor.
• Inaonyesha kwamba model inaweza kufit H(z), supernova na BAO background data kwa accuracy close na ΛCDM.
• Inaonyesha kwamba m na n central values katika combined-data analysis ziko katika preferred dynamical-stability region.
• Inaonyesha kwamba according to AIC, model inaweza kuwa competitive katika combined dataset.

Study haithibitishi nini?

• Haithibitishi kwamba dark energy kwa kweli inatokana only na torsion.
• Haionyeshi kwamba f(T) model ni observationally more correct kuliko general relativity au ΛCDM.
• Haithibitishi kwamba Hubble tension imesuluhishwa.
• Haionyeshi kwamba model inafanikiwa katika cosmological perturbations na structure formation.
• Kwa sababu ya broad parameter uncertainties, haiamui definitively separate physical effects za α, γ na m terms.
• Haionyeshi kwamba critical points zinatimiza all physical density na stability conditions across entire observational posterior.
• Haionyeshi kwamba model imepita Solar System, gravitational-wave au local-gravity tests.
• Haionyeshi kwamba limited combined-data advantage katika AIC inadumu katika stronger Bayesian model comparison.

Mbinu na Matokeo ya Utafiti

Methodological design

Summary ya dynamical-system results

Main cosmological results

General assessment ya results

Strongest result ya study ni kwamba selected f(T) function si only inazalisha late-time acceleration, bali pia inaweza kuonyesha radiation na matter eras kama temporary stages ndani ya dynamical system. Central m na n values katika observational MCMC results pia zinaangukia main stability region inayohitajika kwa sequence hii.

Kwa upande mwingine, significant part ya observational fit inaoverlap almost completely na ΛCDM curves. Some parameters kama γ zina broad na degenerate posteriors. Katika combined analysis, limited advantage ya AIC kwa f(T) model inareverseiwa na stronger penalty ya BIC kwa additional parameters.

Kwa hiyo study inaonyesha kwamba proposed torsion model ni compatible na cosmological background-expansion data na inastahili kuchunguzwa. Ili model iwe physically distinguishable kutoka standard cosmology, inahitaji kutestwa kwa structure growth, cosmological perturbations, gravitational lensing, CMB spectrum na other independent observations.

Maelezo ya Chanzo na Mbinu

Jina kamili la asili la study:Theoretical and Observational Analysis of Cosmological Model in f(T) Gravity

Waandishi na mpangilio: Suraj Kumar Behera, S. A. Kadam, Pratik P. Ray na B. Mishra.

Equal first author: Equal contribution au equal-first-authorship information haijaelezwa.

Corresponding author: Hakuna single corresponding author aliyeelezwa explicitly. Study inatoa email addresses za all four authors kwa separate footnote markers.

Institutional affiliations:

• Suraj Kumar Behera: Department of Mathematics, School of Advanced Sciences, VIT-AP University, Amaravati, Andhra Pradesh, India.
• S. A. Kadam: Centre for Interdisciplinary Studies and Research, D Y Patil International University, Akurdi, Pune, Maharashtra, India.
• Pratik P. Ray: VIT-AP University pamoja na Pacif Institute of Cosmology and Selfology, Sagara, Sambalpur, Odisha, India.
• B. Mishra: Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Telangana, India.

DOI:10.2139/ssrn.6981699.

Journal: Publication katika peer-reviewed journal haijathibitishwa.

Publication platform: SSRN.

Original publisher: Hakuna final peer-reviewed publisher. SSRN record ni preprint-platform record.

Publication year na date: 2026; SSRN index record inatoa date ya 23 Juni 2026.

Source type: Research preprint yenye theoretical dynamical-system analysis, numerical cosmological evolution na observational MCMC parameter estimation.

Peer-review status: Study haijapitia peer review. Hii imeonyeshwa chini ya kila page kwa phrase “Preprint not peer reviewed”.

Official source link:Official SSRN study record.

Version note: Kuna earlier version ya same authors iliyochapishwa kama arXiv:2604.10061v1, yenye title na some numerical values tofauti. Kwa hiyo title, parameter results, tables, figures na statistical comparisons zilizoelezwa hapa zimetolewa only kutoka uploaded SSRN version.

Maelezo haya ya Kituruki yameandaliwa baada ya study kuchunguzwa from beginning to end pamoja na full text, mathematical equations, critical-point tables, phase portraits, cosmological-evolution graphs, MCMC contour diagrams, Hubble na distance-modulus curves, AIC–BIC results, center-manifold appendix na references. Hakuna scientific result iliyoongezwa kutoka external sources; external verification ilitumika only kucheck title, authors, DOI, platform, date na version identity.

Results za study zinaonyesha kwamba selected f(T) model inaweza kuwa compatible na cosmological background-expansion data. Hata hivyo, kwa sababu ya broad parameter degeneracies, unspecified MCMC details, pure-tetrad choice, density conditions katika critical points, some typographical inconsistencies katika text na BIC preference ya ΛCDM, model haipaswi kutathminiwa kama validated alternative kwa general relativity.

Lebo za Meta

f(T) gravity, teleparallel cosmology, DESI DR2, dark energy, cosmological MCMC